Accounting for the Quantum Capacitance of Graphite in Constant Potential Molecular Dynamics Simulations

Molecular dynamics (MD) simulations at a constant electric potential are an essential tool to study electrochemical processes, providing microscopic information on the structural, thermodynamic, and dynamical properties. Despite the numerous advances in the simulation of electrodes, they fail to accurately represent the electronic structure of materials such as graphite. In this work, a simple parameterization method that allows to tune the metallicity of the electrode based on a quantum chemistry calculation of the density of states (DOS) is introduced. As a first illustration, the interface between graphite electrodes and two different liquid electrolytes, an aqueous solution of NaCl and a pure ionic liquid, at different applied potentials are studied. It is shown that the simulations reproduce qualitatively the experimentally-measured capacitance; in particular, they yield a minimum of capacitance at the point of zero charge (PZC), which is due to the quantum capacitance (QC) contribution. An analysis of the structure of the adsorbed liquids allows to understand why the ionic liquid displays a lower capacitance despite its large ionic concentration. In addition to its relevance for the important class of carbonaceous electrodes, this method can be applied to any electrode materials (e.g. 2D materials, conducting polymers, etc), thus enabling molecular simulation studies of complex electrochemical devices in the future.

Accelerating QM/MM simulations of electrochemical interfaces through machine learning of electronic charge densities

A crucial aspect in the simulation of electrochemical interfaces consists in treating the distribution of electronic charge of electrode materials that are put in contact with an electrolyte solution. Recently, it has been shown how a machine-learning method that specifically targets the electronic charge density, also known as SALTED, can be used to predict the long-range response of metal electrodes in model electrochemical cells. In this work, we provide a full integration of SALTED with MetalWalls, a program for performing classical simulations of electrochemical systems. We do so by deriving a spherical harmonics extension of the Ewald summation method, which allows us to efficiently compute the electric field originated by the predicted electrode charge distribution. We show how to use this method to drive the molecular dynamics of an aqueous electrolyte solution under the quantum electric field of a gold electrode, which is matched to the accuracy of density-functional theory. Notably, we find that the resulting atomic forces present a small error of the order of 1 meV/Å, demonstrating the great effectiveness of adopting an electron-density path in predicting the electrostatics of the system. Upon running the data-driven dynamics over about 3 ns, we observe qualitative differences in the interfacial distribution of the electrolyte with respect to the results of a classical simulation. By greatly accelerating quantum-mechanics/molecular-mechanics approaches applied to electrochemical systems, our method opens the door to nanosecond timescales in the accurate atomistic description of the electrical double layer.

Impedance of nanocapacitors from molecular simulations to understand the dynamics of confined electrolytes

Nanoelectrochemical devices have become a promising candidate technology across various applications, including sensing and energy storage, and provide new platforms for studying fundamental properties of electrode/electrolyte interfaces. In this work, we employ constant-potential molecular dynamics simulations to investigate the impedance of gold-aqueous electrolyte nanocapacitors, exploiting a recently introduced fluctuation-dissipation relation. In particular, we relate the frequency-dependent impedance of these nanocapacitors to the complex conductivity of the bulk electrolyte in different regimes, and use this connection to design simple but accurate equivalent circuit models. We show that the electrode/electrolyte interfacial contribution is essentially capacitive and that the electrolyte response is bulk-like even when the interelectrode distance is only a few nanometers, provided that the latter is sufficiently large compared to the Debye screening length. We extensively compare our simulation results with spectroscopy experiments and predictions from analytical theories. In contrast to experiments, direct access in simulations to the ionic and solvent contributions to the polarization allows us to highlight their significant and persistent anticorrelation and to investigate the microscopic origin of the timescales observed in the impedance spectrum. This work opens avenues for the molecular interpretation of impedance measurements, and offers valuable contributions for future developments of accurate coarse-grained representations of confined electrolytes.

Charging and discharging a supercapacitor in molecular simulations

As the world moves more toward unpredictable renewable energy sources, better energy storage devices are required. Supercapacitors are a promising technology to meet the demand for short-term, high-power energy storage. Clearly, understanding their charging and discharging behaviors is essential to improving the technology. Molecular Dynamics (MD) simulations provide microscopic insights into the complex interplay between the dynamics of the ions in the electrolyte and the evolution of the charge distributions on the electrodes. Traditional MD simulations of (dis)charging supercapacitors impose a pre-determined evolving voltage difference between the electrodes, using the Constant Potential Method (CPM). Here, we present an alternative method that explicitly simulates the charge flow to and from the electrodes. For a disconnected capacitor, i.e., an open circuit, the charges are allowed to redistribute within each electrode while the sum charges on both electrodes remain constant. We demonstrate, for a model capacitor containing an aqueous salt solution, that this method recovers the charge-potential curve of CPM simulations. The equilibrium voltage fluctuations are related to the differential capacitance. We next simulate a closed circuit by introducing equations of motion for the sum charges, by explicitly accounting for the external circuit element(s). Charging and discharging of the model supercapacitor via a resistance proceed by double exponential processes, supplementing the usual time scale set by the electrolyte dynamics with a novel time scale set by the external circuit. Finally, we propose a simple equivalent circuit that reproduces the main characteristics of this supercapacitor.

A simple efficient algorithm for molecular simulations of constant potential electrodes

Increasingly, society requires high power, high energy storage devices for applications ranging from electric vehicles to buffers on the electric grid. Supercapacitors are a promising contribution to meeting these demands, though there still remain unsolved practical problems. Molecular dynamics simulations can shed light on the relevant molecular level processes in electric double layer capacitors, but these simulations are computationally very demanding. Our focus here is on the algorithmic complexity of the constant potential method (CPM), which uses dedicated electrostatics solvers to maintain a fixed potential difference between two conducting electrodes. We show how any standard electrostatics solver-capable of calculating the energies and forces on all atoms-can be used to implement CPM with a minimum of coding. As an example, we compare our generalized implementation of CPM, based on invocations of the particle-particle-particle-mesh routine of the Large-scale Atomic/Molecular Massively Parallel Simulator, with a traditional implementation based on a dedicated re-implementation of Ewald summation. Both methods yield comparable results on four test systems, with the former achieving a substantial gain in speed and improved scalability. The step from dedicated electrostatic solvers to generic routines is made possible by noting that CPM's traditional narrow Gaussian point-spread of atomic charges on the electrodes effectively endows point-like atoms with chemical hardness, i.e., an intra-atomic energy quadratic in the charge.

Surface polarization enhances ionic transport and correlations in electrolyte solutions nanoconfined by conductors

Layered materials that perform mixed electron and ion transport are promising for energy harvesting, water desalination, and bioinspired functionalities. These functionalities depend on the interaction between ionic and electronic charges on the surface of materials. Here we investigate ion transport by an external electric field in an electrolyte solution confined in slit-like channels formed by two surfaces separated by distances that fit only a few water layers. We study different electrolyte solutions containing monovalent, divalent, and trivalent cations, and we consider walls made of non-polarizable surfaces and conductors. We show that considering the surface polarization of the confining surfaces can result in a significant increase in ionic conduction. The ionic conductivity is increased because the conductors' screening of electrostatic interactions enhances ionic correlations, leading to faster collective transport within the slit. While important, the change in water's dielectric constant in confinement is not enough to explain the enhancement of ion transport in polarizable slit-like channels.

Electrical noise in electrolytes: a theoretical perspective

Seemingly unrelated experiments such as electrolyte transport through nanotubes, nano-scale electrochemistry, NMR relaxometry and surface force balance measurements, all probe electrical fluctuations: of the electric current, the charge and polarization, the field gradient (for quadrupolar nuclei) and the coupled mass/charge densities. The fluctuations of such various observables arise from the same underlying microscopic dynamics of the ions and solvent molecules. In principle, the relevant length and time scales of these dynamics are encoded in the dynamic structure factors. However, modelling the latter for frequencies and wavevectors spanning many orders of magnitude remains a great challenge to interpret the experiments in terms of physical processes such as solvation dynamics, diffusion, electrostatic and hydrodynamic interactions between ions, interactions with solid surfaces, etc. Here, we highlight the central role of the charge-charge dynamic structure factor in the fluctuations of electrical observables in electrolytes and offer a unifying perspective over a variety of complementary experiments. We further analyze this quantity in the special case of an aqueous NaCl electrolyte, using simulations with explicit ions and an explicit or implicit solvent. We discuss the ability of the standard Poisson-Nernst-Planck theory to capture the simulation results, and how the predictions can be improved. We finally discuss the contributions of ions and water to the total charge fluctuations. This work illustrates an ongoing effort towards a comprehensive understanding of electrical fluctuations in bulk and confined electrolytes, in order to enable experimentalists to decipher the microscopic properties encoded in the measured electrical noise.

Structural Evolution Governs Reversible Heat Generation in Electrical Double Layers

Electrical double layer (EDL) formation determines the reversible heat generation of supercapacitors. While classical theories suggest an exothermic nature, experiments revealed that it can be endothermic, depending on the polarization and electrolyte. Here, we perform constant-potential molecular dynamics simulations and develop a lattice gas model to explore the reversible heat of EDL formation in aqueous and ionic liquid (IL) electrolytes. Our Letter reveals that EDL formation in aqueous electrolytes exhibits endothermicity under negative polarization; it shows new complexity of endothermicity followed by exothermicity in ILs, regardless of electrode polarity. These thermal behaviors are determined by the structural evolution during EDL formation, dominated by adsorbed solvent molecules rather than ions in aqueous electrolytes but governed by "demixing" and "vacancy occupation" phenomena in ILs. This Letter provides new insights into the reversible heat of supercapacitors and presents a theoretical approach to investigating thermal behaviors involving the dynamics of EDLs.

PiNNwall: Heterogeneous Electrode Models from Integrating Machine Learning and Atomistic Simulation

Electrochemical energy storage always involves the capacitive process. The prevailing electrode model used in the molecular simulation of polarizable electrode-electrolyte systems is the Siepmann-Sprik model developed for perfect metal electrodes. This model has been recently extended to study the metallicity in the electrode by including the Thomas-Fermi screening length. Nevertheless, a further extension to heterogeneous electrode models requires introducing chemical specificity, which does not have any analytical recipes. Here, we address this challenge by integrating the atomistic machine learning code (PiNN) for generating the base charge and response kernel and the classical molecular dynamics code (MetalWalls) dedicated to the modeling of electrochemical systems, and this leads to the development of the PiNNwall interface. Apart from the cases of chemically doped graphene and graphene oxide electrodes as shown in this study, the PiNNwall interface also allows us to probe polarized oxide surfaces in which both the proton charge and the electronic charge can coexist. Therefore, this work opens the door for modeling heterogeneous and complex electrode materials often found in energy storage systems.

Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations

Using Brownian dynamics simulations, we investigate the effects of confinement, adsorption on surfaces, and ion-ion interactions on the response of confined electrolyte solutions to oscillating electric fields in the direction perpendicular to the confining walls. Nonequilibrium simulations allows to characterize the transitions between linear and nonlinear regimes when varying the magnitude and frequency of the applied field, but the linear response, characterized by the frequency-dependent conductivity, is more efficiently predicted from the equilibrium current fluctuations. To that end, we (rederive and) use the Green-Kubo relation appropriate for overdamped dynamics, which differs from the standard one for Newtonian or underdamped Langevin dynamics. This expression highlights the contributions of the underlying Brownian fluctuations and of the interactions of the particles between them and with external potentials. Although already known in the literature, this relation has rarely been used to date, beyond the static limit to determine the effective diffusion coefficient or the DC conductivity. The frequency-dependent conductivity always decays from a bulk-like behavior at high frequency to a vanishing conductivity at low frequency due to the confinement of the charge carriers by the walls. We discuss the characteristic features of the crossover between the two regimes, most importantly how the crossover frequency depends on the confining distance and the salt concentration, and the fact that adsorption on the walls may lead to significant changes both at high and low frequencies. Conversely, our results illustrate the possibility to obtain information on diffusion between walls, charge relaxation, and adsorption by analyzing the frequency-dependent conductivity.

Frequency-Dependent Impedance of Nanocapacitors from Electrode Charge Fluctuations as a Probe of Electrolyte Dynamics

The frequency-dependent impedance is a fundamental property of electrical components. We show that it can be determined from the equilibrium dynamical fluctuations of the electrode charge in constant-potential molecular simulations, extending in particular a fluctuation-dissipation relation for the capacitance recovered in the low-frequency limit and provide an illustration on water-gold nanocapacitors. This Letter opens the way to the interpretation of electrochemical impedance measurements in terms of microscopic mechanisms, directly from the dynamics of the electrolyte, or indirectly via equivalent circuit models as in experiments.

Co-Ion Desorption as the Main Charging Mechanism in Metallic 1T-MoS2 Supercapacitors

Metallic 1T-MoS₂ is a promising electrode material for supercapacitor applications. Its layered structure allows the efficient intercalation of ions, leading to experimental volumetric capacitance as high as 140 F/cm³. Molecular dynamics could in principle be used to characterize its charging mechanism; however, unlike conventional nanoporous carbon, 1T-MoS₂ is a multicomponent electrode. The Mo and S atoms have very different electronegativities so that 1T-MoS₂ cannot be simulated accurately using the conventional constant potential method. In this work, we show that controlling the electrochemical potential of the atoms allows one to recover average partial charges for the elements in agreement with electronic structure calculations for the material at rest, without compromising the ability to simulate systems under an applied voltage. The simulations yield volumetric capacitances in agreement with experiments. We show that due to the large electronegativity of S, the co-ion desorption is the main charging mechanism at play during the charging process. This contrasts drastically with carbon materials for which ion exchange and counterion adsorption usually dominate. In the future, our method can be extended to the study of a wide range of families of 2D layered materials such as MXenes.

Effects of surface rigidity and metallicity on dielectric properties and ion interactions at aqueous hydrophobic interfaces

Using classical molecular dynamics simulations we investigate the dielectric properties at interfaces of water with graphene, graphite, hexane and water vapor. For graphite we compare metallic and non-metallic versions. At the vapor-liquid water and hexane-water interfaces the laterally averaged dielectric profiles are significantly broadened due to interfacial roughness and only slightly anisotropic. In contrast, at the rigid graphene surface the dielectric profiles are strongly anisotropic and the perpendicular dielectric profile exhibits pronounced oscillations and sign changes. The interfacial dielectric excess, characterized by the shift of the dielectric-dividing-surface with respect to the Gibbs-dividing-surface, is positive for all surfaces, showing that water has an enhanced dielectric response at hydrophobic surfaces. The dielectric-dividing-surface positions vary significantly among the different surfaces, which points to pronounced surface-specific dielectric behavior. The interfacial repulsion of a chloride ion is shown to be dominated by electrostatic interactions for the soft fluid-fluid interfaces and by non-electrostatic Lennard-Jones interactions for the rigid graphene-water interface. A linear tensorial dielectric model for the ion-interface interaction with sharp dielectric interfaces located on the dielectric-dividing-surface positions works well for graphene but fails for vapor and hexane, because these interfaces are smeared out. The repulsion of chloride from the metallic and non-metallic graphite versions differs very little, which reflects the almost identical interfacial water structure and can be understood based on linear continuum dielectric theory. Interface flexibility shows up mostly in the non-linear Coulomb part of the ion-interface interaction, which changes significantly close to the interfaces and signals the breakdown of linear dielectric continuum theory.

Constant-Potential Molecular Dynamics Simulations of Molten-Salt Double Layers for FLiBe and FLiNaK

We report the results of constant-potential molecular dynamics simulations of the double- layer interface between molten FLiBe and FLiNaK fluoride mixtures and idealized solid electrodes. Employing methods similar to those used in studies of chloride double layers, we compute the structure and differential capacitance of molten fluoride electric double layers as a function of applied voltage. The role of molten salt structure is probed through comparisons between FLiBe and FLiNaK, which serve as models for strong and weak associate- forming salts, respectively. In FLiBe, screening involves changes in Be-F-Be angles and alignment of the oligomers parallel to the electrode, while in FLiNaK the electric field is screened mainly by rearrangement of individual ions, predominantly the polarizable potassium cation.

ELECTRODE: An electrochemistry package for atomistic simulations

Constant potential methods (CPM) enable computationally efficient simulations of the solid-liquid interface at conducting electrodes in molecular dynamics (MD). They have been successfully used, for example, to realistically model the behavior of ionic liquids or water-in-salt electrolytes in supercapacitors and batteries. The CPM models conductive electrodes by updating charges of individual electrode atoms according to the applied electric potential and the (time-dependent) local electrolyte structure. Here we present a feature-rich CPM implementation, called ELECTRODE, for the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS), which includes a constrained charge method and a thermo-potentiostat. The ELECTRODE package also contains a finite-field approach, multiple corrections for non-periodic boundary conditions of the particle-particle particle-mesh solver, and a Thomas-Fermi model for using non-ideal metals as electrodes. We demonstrate the capabilities of this implementation for a parallel-plate electrical double-layer capacitor, for which we have investigated the charging times with the different implemented methods and found an interesting relationship between water and ionic dipole relaxations. To prove the validity of the one-dimensional correction for the long-range electrostatics, we estimated the vacuum capacitance of two co-axial carbon nanotubes and compared it to structureless cylinders, for which an analytical expression exists. In summary, the ELECTRODE package enables efficient electrochemical simulations using state-of-the-art methods, allowing one to simulate even heterogeneous electrodes. Moreover, it allows unveiling more rigorously how electrode curvature affects the capacitance with the one-dimensional correction.

Microscopic Simulations of Electrochemical Double-Layer Capacitors

Electrochemical double-layer capacitors (EDLCs) are devices allowing the storage or production of electricity. They function through the adsorption of ions from an electrolyte on high-surface-area electrodes and are characterized by short charging/discharging times and long cycle-life compared to batteries. Microscopic simulations are now widely used to characterize the structural, dynamical, and adsorption properties of these devices, complementing electrochemical experiments and in situ spectroscopic analyses. In this review, we discuss the main families of simulation methods that have been developed and their application to the main family of EDLCs, which include nanoporous carbon electrodes. We focus on the adsorption of organic ions for electricity storage applications as well as aqueous systems in the context of blue energy harvesting and desalination. We finally provide perspectives for further improvement of the predictive power of simulations, in particular for future devices with complex electrode compositions.

Fully periodic, computationally efficient constant potential molecular dynamics simulations of ionic liquid supercapacitors

Molecular dynamics (MD) simulations of complex electrochemical systems, such as ionic liquid supercapacitors, are increasingly including the constant potential method (CPM) to model conductive electrodes at a specified potential difference, but the inclusion of CPM can be computationally expensive. We demonstrate the computational savings available in CPM MD simulations of ionic liquid supercapacitors when the usual non-periodic slab geometry is replaced with fully periodic boundary conditions. We show how a doubled cell approach, previously used in non-CPM MD simulations of charged interfaces, can be used to enable fully periodic CPM MD simulations. Using either a doubled cell approach or a finite field approach previously reported by others, fully periodic CPM MD simulations produce comparable results to the traditional slab geometry simulations with a nearly double speedup in computational time. Indeed, these savings can offset the additional cost of the CPM algorithm, resulting in periodic CPM MD simulations that are computationally competitive with the non-periodic, fixed charge equivalent simulations for the ionic liquid supercapacitors studied here.

Improving the Accuracy of Atomistic Simulations of the Electrochemical Interface

Atomistic simulation of the electrochemical double layer is an ambitious undertaking, requiring quantum mechanical description of electrons, phase space sampling of liquid electrolytes, and equilibration of electrolytes over nanosecond time scales. All models of electrochemistry make different trade-offs in the approximation of electrons and atomic configurations, from the extremes of classical molecular dynamics of a complete interface with point-charge atoms to correlated electronic structure methods of a single electrode configuration with no dynamics or electrolyte. Here, we review the spectrum of simulation techniques suitable for electrochemistry, focusing on the key approximations and accuracy considerations for each technique. We discuss promising approaches, such as enhanced sampling techniques for atomic configurations and computationally efficient beyond density functional theory (DFT) electronic methods, that will push electrochemical simulations beyond the present frontier.

Origin of Enhanced Performance in Nanoporous Electrical Double Layer Capacitors: Insights on Micropore Structure and Electrolyte Composition from Molecular Simulations

We explore the effect of solvation and micropore structure on the energy storage performance of electrical double layer capacitors using constant potential molecular dynamics simulations of realistically modeled nanoporous carbon electrodes and ionic liquid/organic solvent mixtures. We show that the time-dependent charging profiles of electrodes with larger pores reach the plateau regime faster, while the charging time has a nonmonotonic dependence on ion concentration, mirroring the composition dependence of bulk electrolyte conductivity. When the average pore size of the electrode is similar to or slightly larger than the size of a solvated ion, the solvation enhances ion electrosorption into nanopores by disrupting anion-cation coordination and decreasing the barrier to counterion penetration while blocking the co-ions. In these systems, areal capacitance exhibits a significant nonmonotonic dependence on ion concentration, in which capacitance increases with the introduction of solvent in the concentrated regime followed by a decrease with further dilution. This gives rise to a maximum in capacitance at intermediate dilution levels. When pores are significantly larger than solvated ions, capacitance maximum weakens and eventually disappears. These findings provide novel insights on the combined effect of electrolyte composition and electrode pore size on the charging kinetics and equilibrium behavior of realistically modeled electrical double layer capacitors. Generalization of the approach developed here can facilitate the rational optimization of material properties for electrical double layer capacitor applications.

Multi-scale simulation of the adsorption of lithium ion on graphite surface: From quantum Monte Carlo to molecular density functional theory

The structure of the double-layer formed at the surface of carbon electrodes is governed by the interactions between the electrode and the electrolyte species. However, carbon is notoriously difficult to simulate accurately, even with well-established methods such as electronic density functional theory and molecular dynamics. Here, we focus on the important case of a lithium ion in contact with the surface of graphite, and we perform a series of reference quantum Monte Carlo calculations that allow us to benchmark various electronic density functional theory functionals. We then fit an accurate carbon–lithium pair potential, which is used in molecular density functional theory calculations to determine the free energy of the adsorption of the ion on the surface in the presence of water. The adsorption profile in aqueous solution differs markedly from the gas phase results, which emphasize the role of the solvent on the properties of the double-layer.

Elucidating Curvature-Capacitance Relationships in Carbon-Based Supercapacitors

Nanoscale surface curvatures, either convex or concave, strongly influence the charging behavior of supercapacitors. Rationalizing individual influences of electrode atoms to the capacitance is possible by interpreting distinct elements of the charge-charge covariance matrix derived from individual charge variations of the electrode atoms. An ionic liquid solvated in acetonitrile and confined between two electrodes, each consisting of three undulated graphene layers, serves as a demonstrator to illustrate pronounced and nontrivial features of the capacitance with respect to the electrode curvature. In addition, the applied voltage determines whether a convex or concave surface contributes to increased capacitance. While at lower voltages capacitance variations are in general correlated with ion number density variations in the double layer formed in the concave region of the electrode, for certain electrode designs a surprisingly strong contribution of the convex part to the differential capacitance is found both at higher and lower voltages.

Interfacial water asymmetry at ideal electrochemical interfaces

Controlling electrochemical reactivity requires a detailed understanding of the charging behavior and thermodynamics of the electrochemical interface. Experiments can independently probe the overall charge response of the electrochemical double layer by capacitance measurements and the thermodynamics of the inner layer with potential of maximum entropy measurements. Relating these properties by computational modeling of the electrochemical interface has so far been challenging due to the low accuracy of classical molecular dynamics (MD) for capacitance and the limited time and length scales of ab initio MD. Here, we combine large ensembles of long-time-scale classical MD simulations with charge response from electronic density functional theory to predict the potential-dependent capacitance of a family of ideal aqueous electrochemical interfaces with different peak capacitances. We show that while the potential of maximum capacitance varies, this entire family exhibits an electrode charge of maximum capacitance (CMC) between -2.9 and -2.2 μC/cm², regardless of the details in the electronic response. Simulated heating of the same interfaces reveals that the entropy peaks at a charge of maximum entropy (CME) of -5.1 ± 0.6 μC/cm², in agreement with experimental findings for metallic electrodes. The CME and CMC both indicate asymmetric response of interfacial water that is stronger for negatively charged electrodes, while the difference between CME and CMC illustrates the richness in behavior of even the ideal electrochemical interface.

Microscopic origin of the effect of substrate metallicity on interfacial free energies

We investigate the effect of the metallic character of solid substrates on solid-liquid interfacial thermodynamics using molecular simulations. Building on the recent development of a semiclassical Thomas-Fermi model to tune the metallicity in classical molecular dynamics simulations, we introduce a thermodynamic integration framework to compute the evolution of the interfacial free energy as a function of the Thomas-Fermi screening length. We validate this approach against analytical results for empty capacitors and by comparing the predictions in the presence of an electrolyte with values determined from the contact angle of droplets on the surface. The general expression derived in this work highlights the role of the charge distribution within the metal. We further propose a simple model to interpret the evolution of the interfacial free energy with voltage and Thomas-Fermi length, which allows us to identify the charge correlations within the metal as the microscopic origin of the evolution of the interfacial free energy with the metallic character of the substrate. This methodology opens the door to the molecular-scale study of the effect of the metallic character of the substrate on confinement-induced transitions in ionic systems, as reported in recent atomic force microscopy and surface force apparatus experiments.

A molecular perspective on induced charges on a metallic surface

Understanding the response of the surface of metallic solids to external electric field sources is crucial to characterize electrode-electrolyte interfaces. Continuum electrostatics offer a simple description of the induced charge density at the electrode surface. However, such a simple description does not take into account features related to the atomic structure of the solid and to the molecular nature of the solvent and of the dissolved ions. In order to illustrate such effects and assess the ability of continuum electrostatics to describe the induced charge distribution, we investigate the behavior of a gold electrode interacting with sodium or chloride ions fixed at various positions, in a vacuum or in water, using all-atom constant-potential classical molecular dynamics simulations. Our analysis highlights important similarities between the two approaches, especially under vacuum conditions and when the ion is sufficiently far from the surface, as well as some limitations of the continuum description, namely, neglecting the charges induced by the adsorbed solvent molecules and the screening effect of the solvent when the ion is close to the surface. While the detailed features of the charge distribution are system-specific, we expect some of our generic conclusions on the induced charge density to hold for other ions, solvents, and electrode surfaces. Beyond this particular case, the present study also illustrates the relevance of such molecular simulations to serve as a reference for the design of improved implicit solvent models of electrode-electrolyte interfaces.

Bottom-Up Design of Configurable Oligomer-Derived Conducting Metallopolymers for High-Power Electrochemical Energy Storage

In this Perspective, we sketch out a vision of fast charging and self-healable energy systems that are primarily organic, feature only abundant elements, and operate with ions other than lithium. Using conductive oligomers as highly configurable building blocks, it is possible to create intrinsically adaptable conductive polymeric networks that can be rejuvenated and recycled using simple and safe chemical treatments. Using the versatile organic chemistry toolbox, these oligomers can be further functionalized, for example, with redox-active side chains for high charge storage capacity and ligands capable of complexing metal centers. Cross-linking with metal ions converts the soluble oligomers into insoluble supramolecular networks to yield high-performing electrode materials. The oligomer-based approach can thus provide an exceptional level of control to the design of organic-based battery materials.

Osmotic Transport at the Aqueous Graphene and hBN Interfaces: Scaling Laws from a Unified, First-Principles Description

Osmotic transport in nanoconfined aqueous electrolytes provides alternative venues for water desalination and "blue energy" harvesting. The osmotic response of nanofluidic systems is controlled by the interfacial structure of water and electrolyte solutions in the so-called electrical double layer (EDL), but a molecular-level picture of the EDL is to a large extent still lacking. Particularly, the role of the electronic structure has not been considered in the description of electrolyte/surface interactions. Here, we report enhanced sampling simulations based on ab initio molecular dynamics, aiming at unravelling the free energy of prototypical ions adsorbed at the aqueous graphene and hBN interfaces, and its consequences on nanofluidic osmotic transport. Specifically, we predicted the zeta potential, the diffusio-osmotic mobility, and the diffusio-osmotic conductivity for a wide range of salt concentrations from the ab initio water and ion spatial distributions through an analytical framework based on Stokes equation and a modified Poisson-Boltzmann equation. We observed concentration-dependent scaling laws, together with dramatic differences in osmotic transport between the two interfaces, including diffusio-osmotic flow and current reversal on hBN but not on graphene. We could rationalize the results for the three osmotic responses with a simple model based on characteristic length scales for ion and water adsorption at the surface, which are quite different on graphene and on hBN. Our work provides fundamental insights into the structure and osmotic transport of aqueous electrolytes on 2D materials and explores alternative pathways for efficient water desalination and osmotic energy conversion.

Multifunctional Electrocatalysis on Single-Site Metal Catalysts: A Computational Perspective

Multifunctional electrocatalysts are vastly sought for their applications in water splitting electrolyzers, metal-air batteries, and regenerative fuel cells because of their ability to catalyze multiple reactions such as hydrogen evolution, oxygen evolution, and oxygen reduction reactions. More specifically, the application of single-atom electrocatalyst in multifunctional catalysis is a promising approach to ensure good atomic efficiency, tunability and additionally benefits simple theoretical treatment. In this review, we provide insights into the variety of single-site metal catalysts and their identification. We also summarize the recent advancements in computational modeling of multifunctional electrocatalysis on single-site catalysts. Furthermore, we explain each modeling step with open-source-based working examples of a standard computational approach.

Constant potential simulations on a mesh

Molecular dynamics simulations in a constant potential ensemble are an increasingly important tool to investigate charging mechanisms in next-generation energy storage devices. We present a highly efficient approach to compute electrostatic interactions in simulations employing a constant potential method (CPM) by introducing a particle-particle particle-mesh solver specifically designed for treating long-range interactions in a CPM. Moreover, we present evidence that a dipole correction term-commonly used in simulations with a slab-like geometry-must be used with caution if it is also to be used within a CPM. It is demonstrated that artifacts arising from the usage of the dipole correction term can be circumvented by enforcing a charge neutrality condition in the evaluation of the electrode charges at a given external potential.

The differential capacitance as a probe for the electric double layer structure and the electrolyte bulk composition

In this work, we theoretically study the differential capacitance of an aqueous electrolyte in contact with a planar electrode, using classical density functional theory, and show how this measurable quantity can be used as a probe to better understand the structure and composition of the electric double layer at play. Specifically, we show how small trace amounts of divalent ions can influence the differential capacitance greatly and also how small ions dominate its behavior for high electrode potentials. In this study, we consider primitive model electrolytes and not only use the standard definition of the differential capacitance but also derive a new expression from mechanical equilibrium in a planar geometry. This expression reveals explicitly that the first layer of ions near the charged surface is key to its understanding. Our insights might be used as a guide in experiments to better understand the electrolyte-electrode interface as well as the (composition of the) bulk electrolyte.

Effect of the metallicity on the capacitance of gold-aqueous sodium chloride interfaces

Electrochemistry experiments have established that the capacitance of electrode-electrolyte interfaces is much larger for good metals, such as gold and platinum, than for carbon-based materials. Despite the development of elaborate electrode interaction potentials, to date molecular dynamics simulations are not able to capture this effect. Here, we show that changing the width of the Gaussian charge distribution used to represent the atomic charges in gold is an effective way to tune its metallicity. Larger Gaussian widths lead to a capacitance of aqueous solutions (pure water and 1 M NaCl) in good agreement with recent ab initio molecular dynamics results. For pure water, the increase in the capacitance is not accompanied by structural changes, while in the presence of salt, the Na⁺ cations tend to adsorb significantly on the surface. For a strongly metallic gold electrode, these ions can even form inner sphere complexes on hollow sites of the surface.

Capacitance and Structure of Electric Double Layers: Comparing Brownian Dynamics and Classical Density Functional Theory

We present a study of the structure and differential capacitance of electric double layers of aqueous electrolytes. We consider electric double layer capacitors (EDLC) composed of spherical cations and anions in a dielectric continuum confined between a planar cathode and anode. The model system includes steric as well as Coulombic ion-ion and ion-electrode interactions. We compare results of computationally expensive, but “exact” , Brownian Dynamics (BD) simulations with approximate, but cheap, calculations based on classical Density Functional Theory (DFT). Excellent overall agreement is found for a large set of system parameters, including variations in concentration, ionic size- and valency-asymmetries, applied voltages and electrode separation, provided the differences between the canonical ensemble of the BD simulations and the grand-canonical ensemble of DFT are properly taken into account. In particular, a careful distinction is made between the differential capacitance $$C_N$$ C N at fixed number of ions and $$C_\mu $$ C μ at fixed ionic chemical potential. Furthermore, we derive and exploit their thermodynamic relations. In the future these relations will also be useful for comparing and contrasting experimental data with theories for supercapactitors and other systems. The quantitative agreement between simulation and theory indicates that the presented DFT is capable of accounting accurately for coupled Coulombic and packing effects. Hence it is a promising candidate to cheaply study room temperature ionic liquids at much lower dielectric constants than that of water.

Dynamically Polarizable Force Fields for Surface Simulations via Multi-output Classification Neural Networks

We present a general procedure to introduce electronic polarization into classical Molecular Dynamics (MD) force fields using a Neural Network (NN) model. We apply this framework to the simulation of a solid-liquid interface where the polarization of the surface is essential to correctly capture the main features of the system. By introducing a multi-input, multi-output NN and treating the surface polarization as a discrete classification problem, we are able to obtain very good accuracy in terms of quality of predictions. Through the definition of a custom loss function we are able to impose a physically motivated constraint within the NN itself making this model extremely versatile, especially in the modeling of different surface charge states. The NN is validated considering the redistribution of electronic charge density within a graphene based electrode in contact with an aqueous electrolyte solution, a system highly relevant to the development of next generation low-cost supercapacitors. We compare the performances of our NN/MD model against Quantum Mechanics/Molecular Dynamics simulations where we obtain a most satisfactory agreement.

Intrinsic fractional noise in nanopores: The effect of reservoirs

Fluctuations affect nanoporous transport in complex and intricate ways, making optimization of the signal-to-noise ratio in artificial designs challenging. Here, we focus on the simplest nanopore system, where non-interacting particles diffuse through a pore separating reservoirs. We find that the concentration difference between both sides (akin to the osmotic pressure drop) exhibits fractional noise in time t with mean square average that grows as t^1/2. This originates from the diffusive exchange of particles from one region to another. We fully rationalize this effect, with particle simulations and analytic solutions. We further infer the parameters (pore radius and pore thickness) that control this exotic behavior. As a consequence, we show that the number of particles within the pore also exhibits fractional noise. Such fractional noise is responsible for noise spectral density scaling as 1/f^3/2 with frequency f, and we quantify its amplitude. Our theoretical approach is applicable to more complex nanoporous systems (for example, with adsorption within the pore) and drastically simplifies both particle simulations and analytic calculus.

Molecular Simulation of Electrode-Solution Interfaces

Many key industrial processes, from electricity production, conversion, and storage to electrocatalysis or electrochemistry in general, rely on physical mechanisms occurring at the interface between a metallic electrode and an electrolyte solution, summarized by the concept of an electric double layer, with the accumulation/depletion of electrons on the metal side and of ions on the liquid side. While electrostatic interactions play an essential role in the structure, thermodynamics, dynamics, and reactivity of electrode-electrolyte interfaces, these properties also crucially depend on the nature of the ions and solvent, as well as that of the metal itself. Such interfaces pose many challenges for modeling because they are a place where quantum chemistry meets statistical physics. In the present review, we explore the recent advances in the description and understanding of electrode-electrolyte interfaces with classical molecular simulations, with a focus on planar interfaces and solvent-based liquids, from pure solvent to water-in-salt electrolytes.

Controlling potential difference between electrodes based on self-consistent-charge density functional tight binding

A proper understanding and description of the electronic response of the electrode surfaces in electrochemical systems are quite important because the interactions between the electrode surface and electrolyte give rise to unique and useful interfacial properties. Atomistic modeling of the electrodes requires not only an accurate description of the electronic response under a constant-potential condition but also computational efficiency in order to deal with systems large enough to investigate the interfacial electrolyte structures. We thus develop a self-consistent-charge density functional tight binding based method to model a pair of electrodes in electrochemical cells under the constant-potential condition. The method is more efficient than the (ab initio) density functional theory calculations so that it can treat systems as large as those studied in classical atomistic simulations. It can also describe the electronic response of electrodes quantum mechanically and more accurately than the classical counterparts. The constant-potential condition is introduced through a Legendre transformation of the electronic energy with respect to the difference in the number of electrons in the two electrodes and their electrochemical potential difference, through which the Kohn-Sham equations for each electrode are variationally derived. The method is applied to platinum electrodes faced parallel to each other under an applied voltage. The electronic response to the voltage and a charged particle is compared with the result of a classical constant-potential method based on the chemical potential equalization principle.

Multi-scale approach for the prediction of atomic scale properties

Electronic nearsightedness is one of the fundamental principles that governs the behavior of condensed matter and supports its description in terms of local entities such as chemical bonds. Locality also underlies the tremendous success of machine-learning schemes that predict quantum mechanical observables - such as the cohesive energy, the electron density, or a variety of response properties - as a sum of atom-centred contributions, based on a short-range representation of atomic environments. One of the main shortcomings of these approaches is their inability to capture physical effects ranging from electrostatic interactions to quantum delocalization, which have a long-range nature. Here we show how to build a multi-scale scheme that combines in the same framework local and non-local information, overcoming such limitations. We show that the simplest version of such features can be put in formal correspondence with a multipole expansion of permanent electrostatics. The data-driven nature of the model construction, however, makes this simple form suitable to tackle also different types of delocalized and collective effects. We present several examples that range from molecular physics to surface science and biophysics, demonstrating the ability of this multi-scale approach to model interactions driven by electrostatics, polarization and dispersion, as well as the cooperative behavior of dielectric response functions.

Mass-zero constrained molecular dynamics for electrode charges in simulations of electrochemical systems

Classical molecular dynamics simulations have recently become a standard tool for the study of electrochemical systems. State-of-the-art approaches represent the electrodes as perfect conductors, modeling their responses to the charge distribution of electrolytes via the so-called fluctuating charge model. These fluctuating charges are additional degrees of freedom that, in a Born-Oppenheimer spirit, adapt instantaneously to changes in the environment to keep each electrode at a constant potential. Here, we show that this model can be treated in the framework of constrained molecular dynamics, leading to a symplectic and time-reversible algorithm for the evolution of all the degrees of freedom of the system. The computational cost and the accuracy of the new method are similar to current alternative implementations of the model. The advantage lies in the accuracy and long term stability guaranteed by the formal properties of the algorithm and in the possibility to systematically introduce additional kinematic conditions of arbitrary number and form. We illustrate the performance of the constrained dynamics approach by enforcing the electroneutrality of the electrodes in a simple capacitor consisting of two graphite electrodes separated by a slab of liquid water.

Adiabatic motion and statistical mechanics via mass-zero constrained dynamics

In recent work [Coretti et al., J. Chem. Phys., 2018, 149, 191102], a new algorithm to solve numerically the dynamics of the shell model for polarization was presented. The approach, broadly applicable to systems involving adiabatically separated dynamical variables, employs constrained molecular dynamics to strictly enforce the condition that the force on the fast degrees of freedom, modeled as having zero mass, is null at each time step. The algorithm is symplectic and fully time reversible, and results in stable and efficient propagation. In this paper we complete the discussion of the mechanics of mass-zero constrained dynamics by showing how to adapt it to problems where the fast degrees of freedom must satisfy additional conditions. This extension includes, in particular, the important case of first principles molecular dynamics. We then consider the statistical mechanics of the mass-zero constrained dynamical system demonstrating that the marginal probability sampled by the dynamics in the physical phase space recovers the form of the Born-Oppenheimer probability density. The effectiveness of the approach and the favorable scaling of the algorithm with system size are illustrated in test calculations of solid Na via orbital-free density functional dynamics.